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Rozikov, Utkir Abdulloevich

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Total publications: 35
Scientific articles: 35
Presentations: 9

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Rozikov, Utkir Abdulloevich

Professor
Doctor of physico-mathematical sciences (2001)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 20.05.1970
E-mail: ,
Keywords: dynamical systems, trajectory theory of dinamical systems, nonlinear operators and processes, random walks in random environment, $p$-adic analysis, the Gibbs measures, lattice models of statistical mechanics.quadratic stochastic operators, simplex, trajectory, Volterra and non-Volterra stochastic operators.
UDC: 512.544.23, 517.2, 517.219, 517.958, 517.98, 517.988.52, 517.998, 519.2, 519.21, 519.248, 530.1
MSC: 28d05, 28d15, 22d20, 22d25, 46l35, 60g99, 60b15, 82b20,92b05,97b10

Subject:

The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. For any normal subgroup of finite index the distribution of elements of the partition into conjugate classes on the Cayley tree is described. The notion of wood of Cayley trees is introduced and its group representation is discribed. The uniqueness of the translation-invariant Gibbs measure for the antiferromagnetic Potts model with an external field is proved. For $\lambda $-models the existence of three of translation-invariant and uncountable number of nontranslation-invariant Gibbs measurs on Cayley tree are proved. For any $\lambda$ formula of the critical temperature $ T_c (\lambda )$ is found. The existence of two translation invariant and uncountable numbers of the nontranslationally invariant extreme Gibbs measures for the Ising model with the compete interactions is roved and their constructive description is given. For Ising model with several spin values, Potts model and inhomogeneous Ising model sufficicy conditions of extremity of the unordered phase are obtained. For inhomogeneous Ising model it is proved that there exist only three periodic Gibbs measures corresponding to any normal subgroup of finite index and uncountable number of nonperiodic Gibbs measures. For $\lambda$-models it is proved that there exist only periodic Gibbs measures with period 2 corresponding to any normal subgroups of finite index. On wood of Cayley trees for inhomogeneous Ising model the periodic Gibbs measures are constructed. Using these measures for inhomegeneous Ising model on Cayley tree a new class of Gibbs measures is constructed. A random walk in a random environment on a class of metric spaces are defined. For random walks in a periodic random environment in case when for unit time the particle can transpose to finite distance and for random walks in any random environment when for unit time the particle can transpose only to the neighbouring points, sufficient conditions of non reflexivity are discribed. For example, we consider $Z^d$, infinite trees, free groups.

Biography

Graduated from Faculty of Mathematics and Mechanics of Samarkand State University in 1993 (department mathematical analysis). Ph. D.  thesis was defended in 1995. D. Sci. thesis was defended in 2001. A list of my works contains more than 100 titles.

   
Main publications:
  • Ganikhodjaev N. N., Rozikov U. A. Description of periodic extreme Gibbs measures of some lattice model on the Cayley tree // Theor. and Math. Phys. 1997. V. 111, No. 1, p. 480–486.
  • Rozikov U. A. Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions // Theor. and Math. Phys. 1997. V. 112, No. 1, p. 929–933.
  • Rozikov U. A. Description of limit Gibbs measures for $\lambda$-models on the Bethe lattic // Siberan Math. Jour. 1998. V. 39, No. 2, p. 373–380.
  • Rozikov U. A. Description uncountable number of Gibbs measures for inhomogeneous Ising model // Theor. and Math. Phys. 1999, V. 118, No. 1, p. 77–84.
  • Ganikhodjaev N. N., Rozikov U. A. On unordered phases of certain models on the Cayley tree // Sbornik: Math. 1999. V. 190, No. 2, p. 193–203.
  • Rozikov U. A. Random walks in random environments on metric groups // Math. Notes. 2000, V. 67, No. 1, p. 103–108.
  • Ganikhodjaev N. N., Rozikov U. A. On disordered phase in the ferromagnetic Potts model on the Bethe lattice // Osaka Jour. of Math. 2000. V. 37, No. 2, p. 373–383.
  • Mukhamedov F. M., Rozikov U. A. The disordered phase of the inhomogeneous Potts model is extremal on the Cayley tree // Theor. and Math. Phys. 2000, V. 124, No. 3, 1202–1210.

http://www.mathnet.ru/eng/person8647
List of publications on Google Scholar
https://zbmath.org/authors/?q=ai:rozikov.utkir-a
Full list of publications: Download file (125 kB)

Publications in Math-Net.Ru
2021
1. U. A. Rozikov, “Thermodynamics of interacting systems of DNA molecules”, TMF, 206:2 (2021),  199–209  mathnet  mathscinet; Theoret. and Math. Phys., 206:2 (2021), 174–184
2018
2. U. A. Rozikov, R. M. Khakimov, F. Kh. Khaidarov, “Extremality of the translation-invariant Gibbs measures for the Potts model on the Cayley tree”, TMF, 196:1 (2018),  117–134  mathnet  mathscinet  elib; Theoret. and Math. Phys., 196:1 (2018), 1043–1058  isi  scopus
2017
3. U. A. Rozikov, Z. T. Tugyonov, “Construction of a set of $p$-adic distributions”, TMF, 193:2 (2017),  333–342  mathnet  elib; Theoret. and Math. Phys., 193:2 (2017), 1694–1702  isi  scopus
4. U. A. Rozikov, F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley tree”, TMF, 191:3 (2017),  503–517  mathnet  mathscinet  elib; Theoret. and Math. Phys., 191:3 (2017), 910–923  isi  scopus
5. U. A. Rozikov, M. M. Rakhmatullaev, “Free energies of the Potts model on a Cayley tree”, TMF, 190:1 (2017),  112–123  mathnet  mathscinet  elib; Theoret. and Math. Phys., 190:1 (2017), 98–108  isi  scopus
2013
6. U. A. Rozikov, R. M. Khakimov, “Periodic Gibbs measures for the Potts model on the Cayley tree”, TMF, 175:2 (2013),  300–312  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 175:2 (2013), 699–709  isi  scopus
7. U. A. Rozikov, O. N. Khakimov, “$p$-Adic Gibbs measures and Markov random fields on countable graphs”, TMF, 175:1 (2013),  84–92  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 175:1 (2013), 518–525  isi  scopus
2012
8. U. A. Rozikov, R. M. Khakimov, “The uniqueness condition for a weakly periodic Gibbs measure for the hard-core model”, TMF, 173:1 (2012),  60–70  mathnet  mathscinet  elib; Theoret. and Math. Phys., 173:1 (2012), 1377–1386  isi  scopus
9. F. M. Mukhamedov, U. A. Rozikov, “A polynomial $p$-adic dynamical system”, TMF, 170:3 (2012),  448–456  mathnet  mathscinet  elib; Theoret. and Math. Phys., 170:3 (2012), 376–383  isi  elib  scopus
2011
10. U. A. Rozikov, G. T. Madgoziev, “Nonuniqueness of a Gibbs measure for a model on the Cayley tree”, TMF, 167:2 (2011),  311–322  mathnet  mathscinet; Theoret. and Math. Phys., 167:2 (2011), 668–679  isi  scopus
2010
11. U. A. Rozikov, F. T. Ishankulov, “Description of $p$-harmonic functions on the Cayley tree”, TMF, 162:2 (2010),  266–274  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 162:2 (2010), 222–229  isi  scopus
2009
12. U. A. Rozikov, A. Zada, “On $l$-Volterra quadratic stochastic operators”, Dokl. Akad. Nauk, 424:2 (2009),  168–170  mathnet  mathscinet  elib; Dokl. Math., 79:1 (2009), 32–34  isi  elib  scopus
13. U. U. Zhamilov, U. A. Rozikov, “The dynamics of strictly non-Volterra quadratic stochastic operators on the 2-simplex”, Mat. Sb., 200:9 (2009),  81–94  mathnet  mathscinet  zmath  elib; Sb. Math., 200:9 (2009), 1339–1351  isi  elib  scopus
14. U. A. Rozikov, M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree”, TMF, 160:3 (2009),  507–516  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 160:3 (2009), 1292–1300  isi  scopus
2008
15. U. A. Rozikov, U. U. Zhamilov, “$F$-Quadratic Stochastic Operators”, Mat. Zametki, 83:4 (2008),  606–612  mathnet  mathscinet  zmath  elib; Math. Notes, 83:4 (2008), 554–559  isi  elib  scopus
16. U. A. Rozikov, Sh. A. Shoyusupov, “Fertile HC models with three states on a Cayley tree”, TMF, 156:3 (2008),  412–424  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 156:3 (2008), 1319–1330  isi  scopus
17. U. A. Rozikov, M. M. Rakhmatullaev, “Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree”, TMF, 156:2 (2008),  292–302  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 156:2 (2008), 1218–1227  isi  scopus
2007
18. G. I. Botirov, U. A. Rozikov, “Potts model with competing interactions on the Cayley tree: The contour method”, TMF, 153:1 (2007),  86–97  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 153:1 (2007), 1423–1433  isi  scopus
2006
19. É. P. Normatov, U. A. Rozikov, “A description of harmonic functions via properties of the group representation of the Cayley tree”, Mat. Zametki, 79:3 (2006),  434–443  mathnet  mathscinet  zmath  elib; Math. Notes, 79:3 (2006), 399–407  isi  scopus
20. N. N. Ganikhodzhaev, U. A. Rozikov, “On quadratic stochastic operators generated by Gibbs distributions”, Regul. Chaotic Dyn., 11:4 (2006),  467–473  mathnet  mathscinet  zmath
21. U. A. Rozikov, Sh. A. Shoyusupov, “Gibbs measures for the SOS model with four states on a Cayley tree”, TMF, 149:1 (2006),  18–31  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 149:1 (2006), 1312–1323  isi  scopus
2003
22. N. N. Ganikhodzhaev, U. A. Rozikov, “Group representation of the Cayley forest and some of its applications”, Izv. RAN. Ser. Mat., 67:1 (2003),  21–32  mathnet  mathscinet  zmath; Izv. Math., 67:1 (2003), 17–27  isi  scopus
23. Kh. A. Nazarov, U. A. Rozikov, “Periodic Gibbs Measures for the Ising Model with Competing Interactions”, TMF, 135:3 (2003),  515–523  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 135:3 (2003), 881–888  isi
2002
24. U. A. Rozikov, “Representability of Trees and Some of Their Applications”, Mat. Zametki, 72:4 (2002),  516–527  mathnet  mathscinet  zmath; Math. Notes, 72:4 (2002), 479–488  isi  scopus
25. A. M. Rakhmatullaev, U. A. Rozikov, “Gibbs Measures and Markov Random Fields with Association $I$”, Mat. Zametki, 72:1 (2002),  94–101  mathnet  mathscinet  zmath; Math. Notes, 72:1 (2002), 83–89  isi  scopus
26. N. N. Ganikhodzhaev, F. M. Mukhamedov, U. A. Rozikov, “$\mathbb {Z}$Existence of a Phase Transition for the Potts $p$-adic Model on the Set $\mathbb {Z}$”, TMF, 130:3 (2002),  500–507  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 130:3 (2002), 425–431  isi
27. U. A. Rozikov, “Countably Periodic Gibbs Measures of the Ising Model on the Cayley Tree”, TMF, 130:1 (2002),  109–118  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 130:1 (2002), 92–100  isi
2001
28. U. A. Rozikov, “Periodic Gibbs measures of the inhomogeneous Ising model on trees”, Uspekhi Mat. Nauk, 56:1(337) (2001),  175–176  mathnet  mathscinet  zmath; Russian Math. Surveys, 56:1 (2001), 172–173  isi  scopus
29. F. M. Mukhamedov, U. A. Rozikov, “Von Neumann algebra corresponding to one phase of the inhomogeneous Potts model on a Cayley tree”, TMF, 126:2 (2001),  206–213  mathnet  mathscinet  zmath  elib; Theoret. and Math. Phys., 126:2 (2001), 169–174  isi
2000
30. U. A. Rozikov, “Random walks in random environments of metric groups”, Mat. Zametki, 67:1 (2000),  129–135  mathnet  mathscinet  zmath; Math. Notes, 67:1 (2000), 103–107  isi
31. F. M. Mukhamedov, U. A. Rozikov, “The disordered phase of the inhomogeneous Potts model is extremal on the Cayley tree”, TMF, 124:3 (2000),  410–418  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 124:3 (2000), 1202–1210  isi
1999
32. N. N. Ganikhodzhaev, U. A. Rozikov, “On unordered phases of certain models on a Cayley tree”, Mat. Sb., 190:2 (1999),  31–42  mathnet  mathscinet  zmath; Sb. Math., 190:2 (1999), 193–203  isi  scopus
33. U. A. Rozikov, “Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous Ising model”, TMF, 118:1 (1999),  95–104  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 118:1 (1999), 77–84  isi
1998
34. U. A. Rozikov, “Description of limit Gibbs measures for $\lambda$-models on Bethe lattices”, Sibirsk. Mat. Zh., 39:2 (1998),  427–435  mathnet  mathscinet  zmath; Siberian Math. J., 39:2 (1998), 373–380  isi
1997
35. U. A. Rozikov, “Partition structures of the Cayley tree and applications for describing periodic Gibbs distributions”, TMF, 112:1 (1997),  170–175  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 112:1 (1997), 929–933  isi
36. N. N. Ganikhodzhaev, U. A. Rozikov, “Discription of periodic extreme Gibbs measures of some lattice models on the Cayley tree”, TMF, 111:1 (1997),  109–117  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 111:1 (1997), 480–486  isi

Presentations in Math-Net.Ru
1. Gradient Gibbs measures
U. A. Rozikov
Functional analysis and its applications
March 4, 2021 09:00
2. Mathematical problems in investigation of dynamics of free and bisexual populations
U. A. Rozikov
Functional analysis and its applications
December 19, 2019 10:30
3. A discrete-time dynamical system and an evolution algebra of mosquito population
U. A. Rozikov
Functional analysis and its applications
November 1, 2018 10:30
4. Markov processes of cubic stochastic matrices: Quadratic stochastic processes
U. A. Rozikov
Functional analysis and its applications
April 26, 2018 10:30
5. Thermodynamics of a set of DNA on trees
U. A. Rozikov
Functional analysis and its applications
January 18, 2018 10:30
6. Слабо периодические меры Гиббса для модели Поттса на дереве Кэли
U. A. Rozikov, M. M. Rahmatullaev
Functional analysis and its applications
February 23, 2017 10:30
7. Слабо периодические меры Гиббса для HC-модели на дереве Кэли
U. A. Rozikov, R. M. Khakimov
Seminar of Laboratory of Theory of Functions "Modern Problems of Complex Analysis"
December 22, 2016 12:00
8. Flow of finite-dimensional algebras
U. A. Rozikov
Functional analysis and its applications
December 22, 2016 10:30
9. Меры Гиббса для HC-моделей на дереве Кэли: обзор результатов
U. A. Rozikov, R. M. Khakimov
Functional analysis and its applications
November 17, 2016 10:30

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